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Full text of "Men Of Mathematics"

HEX OF MATHEMATICS
4Xow it is very remarkable that Jerrard's equation lends it-
self with the greatest ease to this research, and is, in the sense
which we shall explain, susceptible of an actual analytic soh-
tion. For we may indeed conceive the question of the algebraic
solution of equations from a point of view different from that
which for long has been indicated by the solution of equations
of the first four degrees, and to which we are especially com-
mitted.
'Instead of expressing the closely interconnected system of
roots, considered as functions of the coefficients, by a formula
involving many-valued radicals,* we may seek to obtain the
roots expressed separately by as many distinct uniform [one-
valued] functions of auxiliary variables, as in the case of the
third degree. In this case, where the equation
a?3  3# -f 2a = 0
is under discussion, it suffices, as we know, to represent the
coefficient a by the sine of an angle, say A, in order that the
roots be isolated as the following well-determined functions
.A       .   a 4- 2ir       .   A -f 4^
2 sin -, 2 sin :-----, 2 sin---------.
3                3                     3
[Hermite is here recalling the familiar ^trigonometric solution'
of the cubic usually discussed hi the second course of school
algebra. The ^auxiliary variable* is A\ the 'uniform functions*
are here sines.]
*Now it is an entirely similar fact which we have to exhibit
concerning the equation
a?5  x  a = 0.
Only, instead of sines or cosines, it is the elliptic functions
which it is necessary to introduce. .,.'
* For example, as in the simple quadratic a2  a = 0: the roots are
us as 4- Vcft and a? =  Va; the 'many-valuedness* of the radical
involved, here a square root, or irrationality of the second degree,
appears in the double sign, =fc, when we say briefly that the two roots
are Va. The formula giving the three roots of cubic equations involves
the three-valued irrationality j^ which has the three values 1,
i( - l+V - 3), i( - 1
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